ACJC 2026 Differential Equations Summary

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Differential Equations / Summary / Page 1 of 4 SUMMARY: Differential Equations Concept Example 1 Type I: d f ( )d y xx = Method: Integrate both sides with respect to x directly. d f( )d d f ( ) d f ( ) d y xx y x x y x x = = =   Example 1: Solve the differential equation 2 d1 ,d 94 y x x = − where 33 22 x−   and hence find the equation of the curve for which y = 1 when x = 0. Solution: 2 d1 d 94 y x x = − Integrating both sides w.r.t. x: 2 1 d 94 yx x = − 2 2 1 d 32 2 x x =  −  112sin23 x C−=+ ------- (1) Sub y = 1 and x = 0 into (1), 111 sin (0)2 C−=+  1C = 112sin 123 xy − = = + ------(2) Type II: d g( )d y yx = Method: Separate variables and integrate both sides respectively. d g( )d 1 d 1 dg( ) 1 dg( ) y yx yxy yxy = = =   Example 2: Find the general solution of the differential equation d 2d y yx = , 0y , expressing y in terms of x. Solution: d 2d y yx = 1 d 2 dyxy = ln 2y x C=+ 2e xCy += 2e xCy += 2e xyA= , eCA= This is a general solution. It is expressed in terms of c, an arbitrary constant. This is a particular solution. We substitute initial conditions into the general equation to find c.

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Differential Equations / Summary / Page 2 of 4 Type III: d f ( )g( )d y xyx = Method: Separate variables and integrate both sides respectively. d f ( ) g( )d 1 d f ( ) dg( ) y xyx y x xy = = Example 3: Find the general solution of the differential equation 2d d y x xyx−= , expressing y in terms of x. Solution: 2d d y x xyx =+ ( ) 2d 1d y xyx =+ 2 1d 1d y xyx =+ 2 1 dd1 y x xy =+ 12 1tan 2y x C− =+ 21tan 2y x C =+ 

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Differential Equations / Summary / Page 3 of 4 Concept Example 2 Solving DE by Substitution Use given substitution to simplify the DE to either Type I, Type II or Type III first and solve the new DE Example 4: Use the substitution y vx= to solve d 3d yx x yx =+ , 0x . Step 1: Differentiate the given substitution to express d d y x in terms of d d v x . Solution: Differentiate y vx= with respect to x on both sides, x vxvx y d d d d += ------------ (1) Step 2: Substitute into DE, and simplify to obtain DE in two variables v and x. Sub (1) and y vx= into the original D.E. : d 3d vx v x x vxx  + = + xx v 3 d d = Step 3: Solve the reduced DE. 3 d 3ln vx x xC = =+  Step 4: Substitute the original variables back into the final answer. Replace v with y x : 3lny xCx =+ 3 lny x x Cx=+

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Differential Equations / Summary / Page 4 of 4 3 Modelling Questions Question DE formulation • d d x t is proportional to x  d d x kxt = , where k is a constant • d d x t is inversely proportional to x  d d xk tx= , where k is a constant • In general, Rate of change = Rate of increase – Rate of decrease E.g. (a) Rate of change of population size = Birth rate – Death rate (b) Rate of change of volume = Rate of In-flow – Rate of Out-flow [2013 MI/I/9] When a cake is removed from the oven, its temperature decreases at a rate proportional to the positive difference between its temperature and the temperature of the room . The temperature of the room is constant at 25 C and T is the temperature of the cake t hours after removing from the oven. Form the differential equation. d ( 25)d T kTt =− where k is a constant [2010 Promo MI/PU2/I/7] It is given that the population of an insect colony is x (in thousands) at time t (in months). The population increases at a rate that is proportional to its population x at time t. In any month, 3000 insects will die of natural causes. Form the differential equation. d d x t = Birth rate – Death rate d 3d x kxt =− where k is a constant [2012 MI 12/I/11 modified] A rectangular tank has a horizontal square base. Water is flowing into the tank at a constant rate. Water is flowing out of the tank at a rate which is proportional to the depth of water in the tank. At time t seconds, the depth of water in the tank is x metres. Form the differential equation. d d V t = Rate of water inflow − Rate of water outflow a bx=− Since dd Areadd Vx tt = , dArea d x a bxt = − d d x c dxt = − where c and d are constants [TJC 2014/I/6] A contagious disease was found to infect a village with a population of 10000 people. Let x, in thousands, be the number of infected people t days after the start of the outbreak. The disease spread at a rate that was proportional to the product of the number of infected people and the number of non-infected people. Form the differential equation. d (10 )d x k x xt =− where k is a constant